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\title[Matematika 1]
{Matematika 1}

\subtitle{8. vaja}

\author[Borut] % 
{B.~Jur\v{c}i\v{c}~Zlobec\inst{1}}

\institute[Univerza v Ljubljani Slovenia] % 
{
  \inst{1}%
    Univerza v Ljubljani,\\Fakulteta za Elektrotehniko\\ 
    1000 Ljubljana, Tr\v{z}a\v{s}ka 25, Slovenija
}


\date[BF 2012] % 
{Matematika FE, Ljubljana, Slovenija \today}

\subject{Matematika}

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\begin{document}


\begin{frame}
 \titlepage
\end{frame}

%\begin{frame}{}
%  \tableofcontents
%\end{frame}
\section{Uporaba integrala v geometriji}
\subsection{Krivulje v ravnini}
\begin{frame}{Parametri\v{c}na oblika
\formulas{$x=x(t), y=y(t),\quad t\in [t_1,t_2]\subset\mathbb{R}$}.}
	\begin{enumerate}
		\item Funkciji \formulas{$x=x(t)$} in 
			\formulas{$y=y(t)$} sta zvezni
		in odvedljivi definirani na \formulas{$[t_1,t_2]$}.
		\item Smerni koeficient tangente
		\formulas{$\displaystyle\frac{dy}{dx}=
		\frac{\dot{y}(t)}{\dot{x}(t)}$}.
		\item Plo"s"cina zanke 
		\formulas{$\displaystyle S=
		\frac{1}{2}\int_{t_1}^{t_2}\left(x(t)\dot{y}(t)-
		y(t)\dot{x}(t)\right)dt$.}
		\item Dol"zina loka 
		\formulas{$\displaystyle
		s=\int_{t_1}^{t_2}\sqrt{\dot{x}(t)^2+\dot{y}(t)^2}dt$.}
	\end{enumerate}
\end{frame}

\begin{frame}{Eksplicitna oblika
\formulas{$y=f(x),\quad x\in[x_1,x_2]\subset\mathcal{D}_f\subset\mathbb{R}$}.}
	\begin{itemize}
		\item Kot poseben primer parametri\v{c}ne oblike 
		\formulas{$x=x$, $y=f(x)$}.
		\item Smerni koeficient tangente 
		\formulas{$\displaystyle\frac{dy}{dx}=f'(x)$}.
		\item Plo"s"cina zanke, ki jo dolo"ca krivulja in os $x$
		\formulas{
		$\displaystyle S=\int_{x_1}^{x_2}f(x)\,dx$}.
		\item Dol"zina loka 
		\formulas{$\displaystyle s=\int_{x_1}^{x_2}\sqrt{1+f'(x)^2}dx$}.
	\end{itemize}
\end{frame}

\begin{frame}{Polarna oblika
\formulas{$r=r(\varphi),\quad
\varphi\in[\varphi_1,\varphi_2]\subset\mathbb{R}$}}.
        \begin{itemize}
                \item Kot poseben primer parametri\v{c}ne oblike\\ 
                \formulas{$x=r(\varphi)\cos\varphi$}, 
		\formulas{$y=r(\varphi)\sin\varphi$}.
                \item Smerni koeficient tangente 
		\formulas{$\displaystyle\frac{dy}{dx}=
		\frac{r(\varphi)\cos\varphi+r'(\varphi)\sin\varphi}
		{-r(\varphi)\sin\varphi + r'(\varphi)\cos\varphi}$}.
                \item Plo"s"cina obmo"cja, ki jo dolo"ca krivulja in
		kota \formulas{$\varphi_1$}, 
		\formulas{$\varphi_2$} je  
                \formulas{$\displaystyle 
		S=\frac{1}{2}\int_{\varphi_1}^{\varphi_2}
		r(\varphi)^2\,d\varphi$}.
                \item Dol"zina loka med kotoma 
		\formulas{$\varphi_1$} in \formulas{$\varphi_2$} je\\ 
		\formulas{$\displaystyle s=\int_{\varphi_1}^{\varphi_2}
		\sqrt{r(\varphi)^2+r'(\varphi)^2}d\varphi$}.
        \end{itemize}
\end{frame}

\begin{frame}{Implicitna oblika \formulas{$f(x,y)=0$}.}
	\begin{itemize}
		\item Smerni koeficient tangente. Odvajamo implicitno.
		\formulas{$\displaystyle\frac{d}{dx}f(x,y(x))=0$} in
		izrazimo \formulas{$y'(x)$}. 
		\begin{itemize}
		\item Primer $x^2+xy+\frac{y^2}{x}=0,\rightarrow$
		\item $2x+y+xy'+2yy'x+y^2=0,\rightarrow$
		\item \formulas{$\displaystyle y'=-\frac{2x+y+y^2}{x+2xy}$} 
		\end{itemize}
		\item Ve"cinoma bomo imeli opravka s  primeri, ko z uvedbo
        	polarnih koordinat, implicitno obliko prevedemo na polarno.
	\end{itemize}
\end{frame}

\begin{frame}{Graf sr\v{c}nice $(x^2+y^2-x)^2=(x^2+y^2)$}
Vpeljemo polarne koordinate: 
$r^2=x^2+y^2$,\\ $x=r\,\cos\varphi$ in dobimo
\formulas{$r=1+\cos\varphi$}.
\formulas{\bf\tt 
PolarPlot[1+Cos[t],\{t,0,2Pi\},PlotStyle->Thick]
}\\
\begin{center}
	\includegraphics[scale=0.6]{func-57.pdf}
\end{center}
\end{frame}

\begin{frame}{Graf lemniskate $(x^2+y^2)^2=(x^2-y^2)$}
Vpeljemo polarne koordinate in dobimo\\
\formulas{$r=\sqrt{\cos(2\varphi)},\quad 
-\frac{\pi}{4}\le\varphi\le\frac{\pi}{4}$},
\formulas{\bf\tt
PolarPlot[Sqrt[Cos[2t]],\{t,-Pi/4,Pi/4\},\\PlotStyle->Thick]
}\\
\begin{center}
        \includegraphics[scale=0.9]{func-58.pdf}
\end{center}
\end{frame}

\begin{frame}{Graf cikloide $x(t)=t-\sin t,\quad y(t)=1-\cos t$.}
\formulas{\bf\tt
ParametricPlot[{t-Sin[t],1-Cos[t]},\{t,0,2Pi\},\\PlotStyle->Thick]
}\\
\begin{center}
        \includegraphics[scale=0.8]{func-59.pdf}
\end{center}
\end{frame}

\begin{frame}{Graf asteroide $x(t)=\cos^t,\quad y(t)=\sin^3 t$.}
\formulas{\bf\tt
ParametricPlot[{Cos[t]\^{}3,Sin[t]\^{}3},\{t,0,2Pi\},\\PlotStyle->Thick]
}\\
\begin{center}
        \includegraphics[scale=0.6]{func-60.pdf}
\end{center}
\end{frame}


\begin{frame}{Nari"si graf krivulje
\formulas{$\displaystyle y^2=x(1-x)^2$}.}
	\begin{itemize}
		\item Krivulja je sestavljena  iz dveh delov.
		\item $y=\pm\sqrt{x}|1-x|$, $x>0$.
	\end{itemize}
\formulas{\bf\tt
Plot[{Sqrt[x]*Abs[1-x],-Sqrt[x]*Abs[1-x]},{x,0,1.2}];
}
\end{frame}

\begin{frame}{Graf}
\begin{center}
	\includegraphics[scale=0.8]{func-61.pdf}
\end{center}
\end{frame}


\begin{frame}{Nari"si graf krivulje
\formulas{$\displaystyle y^2=\frac{x}{2-x}(1-x)^2$}.}
        \begin{itemize}
                \item Krivulja je sestavljena  iz dveh delov.
                \item $y=\pm\sqrt{\frac{x}{2-x}}|1-x|$, $x>0$.
        \end{itemize}
\formulas{\bf\tt
Plot[{Sqrt[x/(2-x)]*Abs[1-x],-Sqrt[x/(2-x)]*Abs[1-x]},{x,0,2}];
}
\end{frame}

\begin{frame}{Graf}
\begin{center}
        \includegraphics[scale=0.8]{func-62.pdf}
\end{center}
\end{frame}

\subsection{Plo\v{s}\v{c}ine in dol\v{z}ine lokov}
\begin{frame}{Izra"cunaj plo"s"cino zanke
\formulas{$\displaystyle y^2=x(1-x)^2$}.}
\begin{itemize}
	\item $\displaystyle S=2\int_0^1\sqrt{x}(1-x)\,dx\rightarrow$.
	\item $\displaystyle 2\left.\frac{2}{15}x^{3/2}(-5+3x)\right|_0^1$.
	\item \formulas{$\displaystyle S=\frac{4}{15}$}.
\end{itemize}
\end{frame}

\begin{frame}{Izra"cunaj plo"s"cino pod lokom cikloide
\formulas{$x=t-\sin t, y=1-\cos t$}.}
\begin{itemize}
        \item $\displaystyle S=\int_0^{2\pi} y(t) \dot{x}(t)dt,\rightarrow$
        \item $\displaystyle\int_0^{2\pi}(1-\cos t)^2dt,\rightarrow$
	\item $\displaystyle\left.
	\frac{3t}{2}-2\sin t+\frac{1}{4}\sin(2t)\right|_0^{2\pi}$.
        \item \formulas{$S=3\pi$}.
\end{itemize}
\end{frame}

\begin{frame}{Izra"cunaj plo"s"cino sr"cnice
\formulas{$r(\varphi)=1+\cos\varphi$}.}
\begin{itemize}
	\item $\displaystyle S=\frac{1}{2}\int_0^{2\pi}(1+\cos\varphi)^2d\varphi,\rightarrow$
	\item $\displaystyle\int_0^{2\pi}(1+2\cos\varphi+\cos^2\varphi)d\varphi,\rightarrow$
	\item $\displaystyle\left.
	\frac{3t}{2}+2\sin t+\frac{1}{4}\sin(2t)\right|_0^{2\pi}$.
	\item \formulas{$S=3\pi$}.
\end{itemize} 
\end{frame}

\begin{frame}{Izra"cunaj plo"s"cino lemniskate
\formulas{$\displaystyle(x^2+y^2)^2=x^2-y^2$}.}
\begin{itemize}
	\item Polarna oblika $r(\varphi)=\sqrt{\cos(2\varphi)}$.
	\item $\displaystyle2\int_{-\pi/4}^{\pi/4}\cos(2\varphi)d\varphi$.
	\item \formulas{$S=2$}.
\end{itemize} 
\end{frame}


\begin{frame}{Izra"cunaj plo"s"cino asteroide
\formulas{$\displaystyle x^{2/3}-y^{2/3}=1$}.}
\begin{itemize}
        \item Parametri"cna oblika $x(t)=\cos^3t,y(t)=\sin^3t$.
        \item $\displaystyle S=\frac{1}{2}\int_{0}^{2\pi}
		\left(x(t)\dot{y}(t)-y(t)\dot{x}(t)\right)dt\rightarrow$
        \item $\displaystyle 3\int_0^{2\pi}\sin^2t\cos^2t\,dt\rightarrow
		\frac{3}{4}\int_0^{2\pi}\sin^2(2t)\,dt\rightarrow
		\frac{3}{8}\int_0^{2\pi}(1-\cos(4x)\,dx$
	\item $S=\displaystyle 3\left.\frac{t}{8}-
		\frac{1}{32}\sin(4t)\right|_0^{2\pi}$, 
        \formulas{$\displaystyle S=\frac{3\pi}{4}$}.
\end{itemize}
\end{frame}


\begin{frame}{Izra"cunaj dol"zino enega loka cikloide
\formulas{$x(t)=t-\sin t, y(t)=1-\cos t, t\in[0,2\pi]$}.}
\begin{itemize}
	\item $\displaystyle s=\int_0^{2\pi}\sqrt{\dot{x}^2(t)+\dot{y}^2(t)}dt,\rightarrow$
	\item $\displaystyle \int_0^{2\pi}\sqrt{2-2\cos t}dt,\rightarrow$
	\item $\displaystyle \int_0^{2\pi}2|\sin(t/2)|dt=
	\left.4\cos(t/2)\phantom{\frac{}{}}\right|_0^{2\pi}$.
	\item \formulas{$s=8$}.
\end{itemize}
\end{frame}

\begin{frame}{Izra"cunaj dol"zino krivulje
\formulas{$r(\varphi)=\frac{1}{\cos\varphi}, \varphi\in[0,\frac{\pi}{4}]$}.}
\begin{itemize}
	\item $\displaystyle x(\varphi)=r(\varphi)\cos\varphi=1$,     
	\item $\displaystyle y(\varphi)=r(\varphi)\sin\varphi=\tan\varphi$.
	\item $S=\displaystyle\int_0^{\pi/4}(1+\tan^2\varphi)d\varphi=
		\left.\tan\varphi\phantom{\frac{}{}}\right|_0^{\pi/4}\rightarrow$
	\formulas{$S=1$}.
\end{itemize}
\end{frame} 

\subsection{Vrtenine}
\begin{frame}{Prostornina in povr\v{s}ina
vrtenine, parametri"cna oblika.} 
Vrtenina je dobljena z vrtenjem 
krivulje \formulas{$y=f(x)$} okoli osi 
$x$ za  \formulas{$x_1\le x \le x_2$}. 
	\begin{itemize}
		\item Prostornina 
			\formulas{$$V=\pi\int_{x_1}^{x_2}f(x)^2 dx$$}
		\item Povr"sina 			
			\formulas{$$ P=2\pi\int_{x_1}^{x_2}f(x)
			\sqrt{1+f'(x)^2}\,dx$$}
	\end{itemize}
\end{frame}

\begin{frame}{Prostornina in povr\v{s}ina
vrtenine, parametri"cna oblika.} 
Vrtenina je dobljena z vrtenjem 
krivulje \formulas{$x=x(t)$, $y=y(t)$} okoli osi 
$x$ za  \formulas{$t_1\le t \le t_2$}. 
	\begin{itemize}
		\item Prostornina 
			\formulas{$$V=\pi\int_{t_1}^{t_2}y(t)^2\dot{x}(t)\,dt$$}
		\item Povr"sina 			
			\formulas{$$ P=2\pi\int_{t_1}^{t_2}y(t)
			\sqrt{\dot{x}(t)^2+\dot{y}(t)^2}\,dt$$}
	\end{itemize}
\end{frame}



\end{document}



